T-7

Goal: the goal of this set of problems is to derive an estimate for the transition point for the Anderson model on the Bethe lattice.
Techniques: cavity method, stability analysis.

A criterion for localization

Green functions and self-energies. Given a lattice with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N } sites $a$ , we call $|a\rangle$ the wave function completely localised in site $a$ . The Anderson model has Hamiltonian
$H=W\sum _{a}\epsilon _{a}|a\rangle \langle a|+\sum _{<a,b>}V_{ab}\left(|a\rangle \langle b|+|b\rangle \langle a|\right)=H_{0}+V$

where the local fields $\epsilon _{a}$ are random variables. We introduce the Green functions $G_{ab}(z)$ and the local self-energies $\sigma _{a}(z)$ : these are functions of a complex variable belonging to the upper half of the complex plane, and are defined by [NOTA SU STILTJIES]

$G_{ab}(z)=\langle a|{\frac {1}{z-H}}|b\rangle ,\quad \quad G_{aa}(z)=\langle a|{\frac {1}{z-H}}|a\rangle ={\frac {1}{z-\epsilon _{a}-\sigma _{a}(z)}}.$

It is clear that when the kinetic term Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V } in the Hamiltonian vanishes, the local self-energies vanish; these quantities encode how much the energy levels Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_a } are shifted by the presence of the kinetic term $V$ . They are random functions, because the Hamiltonian contains randomness. They encode properties on the spectrum of the Hamiltonian; the local density of eigenvalues Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_{a, N}(E)} for an Hamiltonian of size Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N } is in fact given by
$\rho _{a,N}(E)=-{\frac {1}{\pi }}\lim _{\eta \to 0}\Im G_{aa}(E+i\eta )=\sum _{\alpha =1}^{N}|\langle E_{\alpha }|a\rangle |^{2}\delta (E-E_{\alpha }),$
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_\alpha } are the eigenvalues of the Hamiltonian. [NOTA SU PLEMELJI]

A criterion for localization. The local self-energies encode some information on whether localization occurs. More precisely, one can claim [CITE] that localization occurs whenever the imaginary part of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma(E+ i\eta)} goes to zero when $\eta \to 0$ . Given the randomness, this criterion should however be formulated probabilistically. One has:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{\eta \to 0} \lim_{N \to \infty} \mathbb{P}\left(- \Im \sigma_a(E+i \eta)>0 \right)=0 \quad \Longrightarrow \quad \text{Localization} }